On polymodular spacesAbstract. So-called polymodular pseudotopologies and spaces are defined. Lattices of

AN NALES SOCIETATIS M A T H E M A TIC A E PO LO N A E Series I: C O M M E N T A T IO N E S M A T H E M A TIC A E X X IX (1989) R O C Z N IK I P O L S K IE G O T O W A R ZY S T W A M A T E M A T Y C ZN E G O

Séria I: PRACE M A T E M A T Y C ZN E X X IX (1989)

Ed w a r d Am b r o z k o (Poznan)

On polymodular spaces

Abstract. So-called polymodular pseudotopologies and spaces are defined. Lattices of polymodular pseudotopologies are investigated. A certain representation of these pseudotopologies is given. Some theorems on classes of continuous mappings are proved; the theorems can be applied when discussing the polymodular spaces.

The article is a continuation of papers [9], [10], [11], [1] and [2]. Here we shall use many notions and symbols of [1] and [2].

Let us suppose that X is a linear space over the field К of real or complex numbers.

1. Polymodular pseudotopologies and spaces.

1.1. Let у be a positive number. We say that a linear pseudotopology ([5], [6], [7]) t on X (and the space (X, t)) is polymodular of the character у if for every ^ет(О) there exists a filter ^ei(O) n FMy(X) such that ^ c:

Recall that the symbol FMy(X) denotes the set. of all modular filters of the character у defined in X ([2]).

1.1.1. If a pseudotopology т eLP(X) (the set of all linear pseudotopologies on X) is polymodular of the character 1 (or of a character ye(0, 2)), then we say that т is a poly-LT-pseudotopology and (X , t) is a poly-LT-pseudo, topological space (shortly; a poly-LT-space).

1.1.2. Suppose that a pseudotopology % gLP(X) is modular of a character у > 0. Then this pseudotopology is polymodular of the character y.

Proof. Let t = ijr (see [2]), where & e F My(X). Then

t(0) — \fSeF(X): 3 2#" with some number X ф 0}.

Observe that for each ХеК, X ф 0, we have X^er(0) n FMy(X) ([2], 3.1.5).

1.1.3. Let у > 0. We denote by PpMy{X) the set of all linear pseudo­

topologies on X which are polymodular of the character y. Moreover, we define PpLT{X): = PpMf X) .

1.1.4. Let y > 0 and suppose that {ту j e J } c P p M y(X), J # 0 . Then i n W ) ( v J eJi> SUPLP(X)bj- j e J } e P p M y(X).

Proof. Let т := infiPW{r7-: JeJ). Take a filter ^ ex(0) and choose SF x, ^ ne\J Tj(0) such that J* r> 3F x + ... -f ([2], 5). Next, let filters

j e J

Ух, ..., e((J тДО))n FMy(X) satisfy the conditions x о , . . . , & n

H J

з ^ n. Then we have => + ... + ^ „ e i(0) n FMy(X) ([2], 3.1.3). Hence

t ePpMy(X).

Now let a: = supLP(x){Tf 7’ eJ}. Consider a filter e <т(0) = П ^ (0) ( [2],

H J

5). Then for each index j e J there exists a filter e тДО) n FMy(X) such that 3 Thus we have #■ => supPW{#y. je J ) and ^ : = s u p ^ l# ]-: je ./}

e F M y(X) ([2], 3.1.1). Since F(X) э У => J^-етДО) for j e J, the filter ^ belongs to 0 , ( 0 ) = <7(0). Finally, => ^e<r(0) n FMy(X). We have proved that OEPpMy{X).

1.1.5. Give the following corollary:

For each y > 0 the set PpMy(X) is a complete lattice ([3], [8]).

1.1.6. Observe that infLPmLP(X) = x{X}ePpMy(X) (y > 0). So tw is the smallest element of PpMy(X).

1.1.7. Let us define t:= supLPW LP(V). It is easy to show that x(0) = {J* eF(X): ^ =э Vxl + ... +Vxn with some xl5 ..., x „ e l } . Now let the space X be infinite-dimensional. Take a filter 3F et(0) and suppose that & => Vxx + ... + Vxn, x x, . . . , xn e X. Of course, the set Ixx + ... + Ix„e

is not absorbent in X. Therefore we see that no filter of t(0) is modular (cf. with [1], 5.4.6). Hence т фРрМу(Х) (у > 0).

1.2. Let us mention some remarks on a certain special case of the inductive limit.

1.2.1. Suppose that (J, ^ ) is a non-empty directed set ([4]). Let for every JeJ a pseudotopology ZjELP{X) be given. Moreover, assume that for all j x, j 2eJ, j x ^ j 2, the relation xjl ^ xj2 is satisfied. Then

(infLP(X) xj)(x) = U TM) = (J Xj(0) + x for x eX.

je J je J je J

Here the pseudotopology infLP(X)Tj is also denoted by indlimi^ (cf. with

je J je J

[6], Appendix 2; see also [12]).

Proof. It is easy to show that t(0):= (J -гДО) satisfies conditions (a), (J3),

je J

(l)-(4) of [1]. Verify, for example, conditions ((3), (1), (3). Let filters 3F, ^ет(О) be

On polymodular spaces 25

given. Then there exist j x, j 2EJ such that ^ e x h(Q), ^ е т72(0). Take a j 0eJ satisfying the conditions j x < j 0, j 2 ^ i 0- Of course, we have i7l(0) c xjo(0) and Tj2(0) c i7o(0). Therefore, ^ e i JO(0) and J* n У, Ж + У Exjo(0) <= т(0) (con­

ditions (P) and (1) are satisfied). Now consider an arbitrary filter Жех(0).

Obviously, where j 0 is a certain element of J. Hence we get УЖ Exjo(0) cz x(0) (t(0) satisfies condition (3)). We now see that x with

t(x): = t(0) + x( = (JтДх)), x eX, is a linear pseudotopology on X. Since

jeJ

тДО) с= t(0), the relation x ^ x { Je J) holds. Therefore, x ^ inftPWTy. On the

jeJ

other hand, the inclusion t(0) <= (infLP(X) t7)(0) is satisfied ([2], 5), so

je J

infLPmr7 ^ x. Finally, we have x — infLP(X)Xj.

je J je J

1.2.2. Let x be a pseudo topology on a set Y Ф0 . We say that x is a Hausdorff pseudotopology if т (х )п ф ) = 0 for all x, у eY, x ф у.

It is easy to show that a pseudotopology xeLP(X) is Hausdorff if and only if for each x eX the condition [х ]е т(0) implies x = 0.

1.2.3. Suppose that the pseudotopologies Xj appearing in 1.2.1 are Hausdorff.

Then indlimi. is also a Hausdorff pseudotopology.

jeJ

Proof. Let [x] e (J тДО), where x eX. Then [x] ет7о(0) for a certain j 0E J.

je J

Therefore x = 0.

1.2.4. Observe that if the pseudotopologies xj appearing in 1.2.1 are modular of a fixed character у > 0, then indlimi^ (= infLP(X)T7) is polymodular

je J je J

of the character у (see 1.1.4 and 1.1.2).

1.2.5. Let у > 0 and let a pseudotopology xEPpMy(X) be given. We shall show that x = indlirntj for certain pseudotopologies XjEPMy(X), JeJ ф 0.

je J

Consider the set

J : = x ( 0 ) n F M y(X).

Of course, J Ф0 . The set J can be directed in the following manner:

Л ^ i 2 if and оп1У if i i 3 h•

Obviously, the relation ^ is reflexive and transitive. Verify the condition:

For arbitrary j l , j 2EJ there exists a j 0eJ such that i i ^ i o and j 2 ^Jo-

Let filtersii,i2eJ be given. By virtue of [2], 3.1.3 we infer that the filter ii + i2 is modular of the character y. Moreover, we have Л + i2ei(0); therefore,

h + h E j - 11 is easy to see th a t./,/, =>jx+ j 2 ( i . e . , / , / ^ /+ /> ) . Let j e J and let tj be the modular pseudotopology generated by the filter j. We have

tj(x) = {ЗФe F(X): ЗФ з Aj with a certain number Я Ф 0} + x, x e X . Observe that if j 1, j 2eJ and / < j 2, then i71 ^ xj2.

Next, show that

( Jt j(0 ) = t( 0 ) . je J

Let a filter 3F e (J тДО) be given. Then J^ei^O), where j 0 is a certain element

je J

of J. So there exists a number Ae K such that 3F з Aj0. Obviously, Aj0e t(0);

therefore 3^ет(0). Now take a filter ^ет(О). Then ^ з j 0, where j 0 is some filter of J. Hence we have ^ е т ;-о(0) с (J -гДО). We now see that

je J

t = indlimij = in/p^ty.

je J je J

1.2.6. "Observe that the following theorem holds:

Suppose that у > 0, теPpMy(X), {ту. j e J } = {<je PMy(X): т ^ cr}-, h ^ h ( j i J 2eJ) tfand only if Tjl ^ Tj2- Then J is a non-empty directed set and T = indlimTy.

je J

2. Theorems on continuous mappings. Let us give some theorems in which inductive limits (defined as in 1.2.1), intima and suprema of linear pseudo­

topologies appear. Of course, these theorems can be used to discuss properties of polymodular spaces (see 1.1.4, 1.2.4 and 1.2.5).

2.1. Suppose that J ф 0 is a directed set, TjEbP{X) for j e J, т : = indlim т j

je J

(so the assumptions of 1.2.1 must be satisfied), (Y, cr) is a non-empty pseudotopological space, C is the set of all continuous mappings of {X, t) to (Y, a), Cj is the set of all continuous mappings of {X, Tj) to (Y, o), j e J.

Then

• c = 0 c i-

je J

Proof. Let a mapping f e C be given. Since the identity mapping i: {X, Tj)-+(X, t) is continuous, the mapping / : {X, iy)->(Y, cr) is also continuous (for each jeJ). Therefore/e p) C}. Now suppose that f e f ] C j and

je J je J

On polymodular spaces ²⁷

take an element x e X and a filter $F ет(х). Since t(x) = (J zj(x), there exists an

je J

index j 0sJ for which ет;-о(х). The assumption / e Q Cj implies that the

j e J

mapping f : (X, zjo)->(Y, cr) is continuous. Therefore f ( ^ ) e o ( f (x)). Hence f e C .

2.2. Observe that if for j e J # 0 a pseudotopology ZjeLP(X) is given,

t: = infLP{X)Tj, (7, a) is a non-empty pseudotopological space, C is the set of all

je J

continuous mapping of (Y, cr) to (X , t) and C} is the set of all continuous mappings of (7, cr) to (X, zj), j e J, then

U G j c C .

je J

2.3. Mention the following simple lemma:

Let (X, cr) and (7, t) be linear-pseudotopological spaces. Suppose that a mapping f of (X, cr) to (7, t) is additive (i.e.,f(x± + x2) = f ( x j + f (x2) for x1?

x2eX) and continuous at a point x 0e X. Then the mapping f : (X, cr)-»(Y, z) is continuous.

2.4. Let (X, a) and (7, z) be linear-pseudotopological spaces and let the pseudotopology о satisfy condition (M) of [1] with a filter 3F. Then a linear mapping f : (X, o)->(Y, i) is continuous if and only if f ( ^ ) et(0).

Proof. Necessity is obvious. Sufficiency. Assume that the linear map­

ping / : (X, cx)-»(7, t) satisfies the condition f ( ^ ) e i(0). We are going to prove that / is continuous. Of course, it is enough to show that / is continuous at 0. Let <$ e o ( 0). Then ^ =э кЗР with some number k. Hence we have f(^)^>f( k^) . Moreover, the equality f ( k ^ ) = kf (^) holds (because / is homogeneous). Since /(J^)e t(0), we obtain kf (^)ez(O). Therefore /(^ ) GT(0).

2.4.1. Let (X, a), (7, t) be linear-pseudotopological spaces satisfying con­

dition (M) with 2F (for (X, cr)) and У (for (7, x)). Then a linear mapping f : (X , cr) -*■ (7, t) is continuous if and only if there exists a number к such that /(# ") =э № .

This is a corollary of the above theorem.

2.5. Suppose that J Ф 0 (i.e., (J, ^) with J ф 0) is a directed set, ZjeLP(X) are given (JeJ), the inductive limit z: = indlim Zj (defined as in 1.2.1) exists, (7, cr)

jeJ

is a linear-pseudotopological space satisfying condition (M), L is the set of all continuous linear mappings of (7, cr) to (X, z) and Lj is the set of all continuous

linear mappings of (Y, <j) to (X , x}), j e J . Then L = { J Lj.

je J

Proof. Obviously, we have {JLjCzL. Let 3F be the fixed filter of

je J

condition (M) for the pseudotopology a. Take a mapping f eh. Then f ( ^) e x ( 0 ) = (J t7(0), and therefore there exists an index j 0eJ such that / ( ^ ) етУо(°)- Hence f e L jo a (J L.} (see 2.4).

je J

2.6. Let the following conditions be satisfied: a non-empty set J is given, TjELP(X) for j e J , г : = infLP(A:) iy, (Y, <т) is a linear-pseudotopological space,

je J

A is the set of all continuous additive mappings of {X,x) to {Y, a) and Aj is (for j e J ) the set of all continuous additive mappings of (X, Xj) to (Y, <x).

Then

л = П Ar

je J

Proof. The inclusion A c= Q A} is obvious. Suppose that f e f') Aj. Then

je J je J

/ : (X , Ty)->(Y, гг) is continuous for each j e J . Let a filter ^ ex (0) be given and let for certain £F x, ..., J ^g (J тДО). Since the mapping

je J

f is additive, we have / ( # / + . . .d -# -,,) = / ( # / ) + . . . + / ( # / ) . Moreover, / : (X, тЛ-ПУ, <t) is continuous (jeJ); therefore/(J*,), . . . , / ( g cr(0), and f ( ^ i ) + ... + /(J % )g(7(0). So we obtain /(# ") = з /(# '1)+ ... +/(J^n)eo-(0).

Hence/(#')G<r(0). By virtue of 2.3 we infer that the mapping/: (X, t)->>(Y, <j) is continuous.

2.7. Suppose that: for j e J # 0 a pseudotopology XjeLP(X) is given, x: = supLpW T-, (T, cr) is a non-empty pseudotopological space, C is the set of all

je J

continuous mappings of (Y, o) to (X, i) and Cj (jeJ) is the set of all continuous mappings of (Y, cr) to (X, xf. Then

c = П c i-

je J

Proof. The inclusion C <=: f ) Cj is obvious. Suppose that f eÇ\Cj. Take

je J je J

an element x e Y and a filter ^ e o ( x ) . Of course, we have f ( ^ r) exj(f(x)) for each j e J . Therefore/(J^) g Q Ty(/(x)) == t( / (x)). Hence f e C .

je J

On polymodular spaces 29

3. Some classes of polymodular pseudotopologies.

3.1. Let s be a positive number.

We say that a pseudotopology т eLP(X) (and the space (X , г)) is poly-s-modular if for every # "gt(0) there exists a filter i(0 ) n F sM(X) (see [2]) such that У c= SF.

Obviously, an s-modular pseudotopology is poly-s-modular.

The set PpsM(X) of all poly-s-modular pseudotopologies on X is a complete lattice.

Every poly-s-modular pseudotopology can be represented as the inductive limit of s-modular pseudotopologies.

3.2. Fix a number s > 0.

We say that т eLP(X) is a locally s-convex polymodular pseudotopology if for each ^ет(О) the condition ^ c: is satisfied with a certain filter

t(0) n FMS_C (X).

Define PpMs.c.(X)

:= { i eLP(X): if ^ et(0), then ^ c= £F for some ^ет(О)n FMS_CXX)}.

The set PpMs_c (X) is a complete lattice (with respect to the relation

<t ^ t if and only if ff(0) э т(0); <т, т ^ePpMs_c (X)).

Namely, the following theorem is true (also for PpsM(X)):

Suppose that H c: PpMs.c (X) and H # 0 . Then mfLP(X)H, supLP(X)H ePpMs_cXX).

If те PpMs.c (X), then t = indlimiy with a certain directed set J ф 0 and

j e J

some tjEPMs.cXX), JeJ.

3.3. We say that a pseudotopology т eLP(X) (and the space (X, t)) is polymodular (of a floating character) if for every ^ et(0) there exists a filter

&ет(0)п (J FMy(X) such that ^ cr .

y > 0

3.3.1. The set PpMiy)(X) of all such pseudotopologies on X is a complete lattice.

Proof. Let [ту. j ^eJ} c PpMM(X), where J Ф 0 . Consider the pseudo­

topology x: = mîLP(X){xу j e J } . We shall show that т EPpMiy)(X). Take a filter ет(0). Then there exist a number y0 > 0 and filters ..., 3F n e((jTj(0))nFMyo(X) such that ^ Y+ . . . +<Fn (see [2], 5 and 1.3.4).

j eJ

Obviously, + ... + ^ „ е т ( 0 ) п Ш уо(1) ([2], 3.1.3, 5). Hence тePpMiyXX).

Now let о be an arbitrary pseudotopology of PpM{y)(X) and choose a filter J^ecr(0)n (J FMy(X). Construct the Orlicz topology <r: = (x^)v = i^^see

у > 0

[1] and [2]). Of course, we have a ^ x^ ^ a. Consider the topology Q- = supLP(x) {à: a g PpMiy)(X)} (= supLT{X]IT(X)).

Since a ^ q for every a e PpMiy)(X) and since g ePp M{y)(X), we see that Q = supPpM(y)(X)PpMM(X) (= infPpM(y)W0). Hence the set PpMM{X) is a com­

plete lattice; for H cr PpM(y)(X) we have

suppPM(y)(x)H = ^ьр{хЛ°еРрМ{у){Х): x ^ <7 for any xeH } (in particular, supPpAf(y)(X)0 = infLP(Ar)PpM(v)(X) = tw ).

3.3.2. If г e PpMiy){X), then there exist a directed set J ф 0 and pseudo­

topologies Xj e (J PMy(X) ( j e J ) such that x = ind lim t..

y > 0 ^{j e J}

3.4. A pseudotopology xeLP(X) is said to be poly-(s)-modular if for every

& ет(0) there is a filter &ex(0)n (J FSM(X) such that У c .

s > 0

The set Pp(s)M(X) of all poly-(s)-modular pseudotopologies on X is a complete lattice.

A poly-(s)-modular pseudotopology can be represented as the inductive limit of s-modular pseudotopologies (s > 0 need not be fixed).

3.5. We say that xeLP(X) is a locally (s)-convex polymodular pseudo­

topology if for every ^ет(О ) there is a filter ^ е т(0)n (J FMS.C(X) such that

s > 0

c

Let the symbol PpM(s).c (X) denote the set of all locally (s)-convex polymodular pseudotopologies on X.

Observe that if H e PpM(s).cXX), H ф 0 , then infLP{X)H e PpMis).c (X).

For every x EPpM(s).c {X) there exist a directed set J ф 0 and an MS-sequence (xy j e J ) of pseudotopologies of (J PMS.C(X) such that x

s > о

= indlimi,-.^ J jeJ

3.6. We have introduced some classes of polymodular pseudotopologies.

Of course, one can investigate relations between them. For example, if ye (0, 2), then PpMy{X) = PpLT(X).

References

[1 ] E. A m b r o z k o , Pseudotopologies fo r modular spaces, Comment. Math. 25 (1985), 189-200.

[2 ] — , On modular filters, this fasc., 1 -2 1 .

[3 ] G . B i r k h o f f , Lattice Theory, M oscow 1984 (in Russian, translated from English).

[4 ] R. E n g e l k i n g , General Topology, W arszawa 1977.

On polymodular spaces ³¹

[5 ] H. R. F is c h e r , Limesraume, M ath. Ann. 137 (1959), 269-3 0 3.

[6 ] A. F r ô l i c h e r , W . B u c h e r , Calculus in Vector Spaces without Norm, M oscow 1970 (in Russian, translated from English); with appendices prepared by V . I. Averbukh and O . G . Smolyanov.

[7 ] W . G à h l e r , Grundstrukturen der Analysis II, Berlin 1978.

[8 ] G . G r â t z e r , General Lattice Theory, M oscow 1982 (in Russian, translated from English).

[9 ] R. L e s n i e w i c z , On generalized modular spaces I, Comment. Math. 18 (1975), 223 -2 4 2.

[1 0 ] — , On generalized modular spaces II, ibidem 18 (1975), 2 43 -2 7 1.

[11 ] — , W . O r l i c z , A N o te on Modular Spaces. X IV , Bull. Acad. Polon. Sci., Sér. Sci. Math.

Astronom. Phys. 22 (1974), 9 15 -9 2 3.

[1 2 ] M . F. S u h in in , On the local invertibility o f a differentiable mapping (in Russian), Uspehi M at.

Nauk 25 (1970), 2 49 -2 5 0.